Many people read it without realizing the joke he put in the first paragraph.

Seldom do I pause to answer criticism of my work and ideas. If I sought
to answer all the criticisms that cross my desk, my secretaries would
have little time for anything other than such correspondence in
the course of the day, and I would have no time for constructive work.

The joke, in case you missed it, is that he doesn't have a big office full of secretaries that manage his correspondence. Like most social activists he's on a shoe-string budget. The people he is replying to, however, probably do have such luxuries.

I often wonder if his intention was to open the letter with a subtle joke or the most polite "F- you" ever delivered in history.

Nearly a year later when he spoke at Drew University (I wasn't born yet, but I would attend that University 20+ years later) his speech also began with a joke. He quoted the great philosopher Bob Hope. I don't want to spoil the joke for you, so you'll have to listen it it yourself here: http://depts.drew.edu/lib/archives/online_exhibits/King/index.html